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There's a little-known function from Data.List called nub. It's particularly good for code golfing.

import Data.List
exactlyOneVowel :: String -> Bool
exactlyOneVowel = (== 1) . length . nub . filter (`elem` "aeiouyAEIOUY")

I believe this satisfies all the constraints of the problem, and is unfortunately O(#characters ^ 2).